In the mathematical theory of
automorphic representation
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset ...
s, a multiplicity-one theorem is a result about the
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
of an
adelic reductive algebraic group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation that has a finite kernel and is a ...
. The multiplicity in question is the number of times a given abstract
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
is realised in a certain space, of
square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
s, given in a concrete way.
A multiplicity one theorem may also refer to a result about the
restriction of a
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
''G'' to a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
''H''. In that context, the pair (''G'', ''H'') is called a strong
Gelfand pair
In mathematics, a Gelfand pair is a pair (''G'', ''K'') consisting of a group ''G'' and a subgroup ''K'' (called an Euler subgroup of ''G'') that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely re ...
.
Definition
Let ''G'' be a reductive algebraic group over a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
''K'' and let A denote the
adele
Adele Laurie Blue Adkins (; born 5 May 1988) is an English singer-songwriter. Regarded as a British cultural icon, icon, she is known for her mezzo-soprano vocals and sentimental songwriting. List of awards and nominations received by Adele, ...
s of ''K''. Let ''Z'' denote the
centre
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of ''G'' and let be a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
unitary character from ''Z''(''K'')\Z(A)
× to C
×. Let ''L''
20(''G''(''K'')/''G''(A), ) denote the
space of cusp forms with central character ω on ''G''(A). This space decomposes into a
direct sum of Hilbert spaces
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, ma ...
:
where the sum is over
irreducible
In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole.
Emergence plays a central role ...
subrepresentation In representation theory, a subrepresentation of a representation (\pi, V) of a group ''G'' is a representation (\pi, _W, W) such that ''W'' is a vector subspace of ''V'' and \pi, _W(g) = \pi(g), _W.
A nonzero finite-dimensional representation alw ...
s and ''m''
are non-negative
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s.
The group of adelic points of ''G'', ''G''(A), is said to satisfy the multiplicity-one property if any
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
irreducible
admissible representation
In mathematics, admissible representations are a well-behaved class of Group representation, representations used in the representation theory of reductive group, reductive Lie groups and locally compact group, locally compact totally disconnected ...
of ''G''(A) occurs with multiplicity at most one in the space of
cusp form In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion.
Introduction
A cusp form is distinguished in the case of modular forms for the modular gr ...
s of central character , i.e. ''m''
is 0 or 1 for all such .
Results
The fact that the
general linear group
In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
, ''GL''(''n''), has the multiplicity-one property was proved by for ''n'' = 2 and independently by and for ''n'' > 2 using the uniqueness of the
Whittaker model
In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as ''GL''2 over a finite or local or global field on a space of functions on the group. It is named afte ...
. Multiplicity-one also holds for
''SL''(2), but not for ''SL''(''n'') for ''n'' > 2 .
Strong multiplicity one theorem
The strong multiplicity one theorem of and states that two cuspidal automorphic representations of the general linear group are isomorphic if their local components are isomorphic for all but a finite number of places.
See also
*
Gan-Gross-Prasad conjecture
References
*
*
*
*
*
*
*{{Citation , last1=Shalika , first1=J. A. , title=The multiplicity one theorem for GL
''n'' , jstor=1971071 , mr=0348047 , year=1974 , journal=
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
, series=Second Series , issn=0003-486X , volume=100 , issue=2 , pages=171–193 , doi=10.2307/1971071
Representation theory of groups
Automorphic forms
Theorems in number theory
Theorems in representation theory